The Head and Tail of the Colored Jones Polynomial for Adequate Knots
Abstract.
We show that the head and tail functions of the colored Jones polynomial of adequate links are the product of head and tail functions of the colored Jones polynomial of alternating links that can be readoff an adequate diagram of the link. We apply this to strengthen a theorem of Kalfagianni, Futer and Purcell on the fiberedness of adequate links.
1. Introduction
For large classes of links , but not for all links, the colored Jones polynomial, a sequence of (Laurent) polynomial link invariants indexed by a natural number , develops a welldefined tail: Up to a common sign change the first coefficients of agree with the first coefficients of for all . This gives rise to a power series with interesting properties. For example, for many knots with small crossing numbers the tail functions are given by products of onevariable specializations of the twovariable Ramanujan theta function [AD11].
The colored Jones polynomial of the mirror image of satisfies If it exist the tail function of is called the head of . It was shown in [DL07, DL06, DT13] that the head and tail functions for alternating links contain geometric information that can be used to give upper and lower bounds for the hyperbolic volume of a nontorus alternating link. In a series of papers and in a book Futer, Kalfagianni and Purcell extended those results to larger and larger classes of links (e.g. [FKP08, FKP13]). It is the goal of this paper to show that one can express the head or tail functions of an adequate link, a large class of links that contain alternating knots, as products of head or tail functions of alternating links that can be readoff an adequate diagram of the link. As a geometric application we strengthen a Theorem of Futer, Kalfagianni and Purcell related to the fiberedness of an adequate link.
Example 1.1.
Let be the mirror image of the nonalternating knot as in Figure 2. Its colored Jones polynomial  up to multiplication with a suitable power for some integers  is given by:
The data was obtained from Dror BarNatan’s Mathematica package KnotTheory [BN11]. Thus the tail of the colored Jones polynomial of the mirror image of the knot is given by the series:
We will study the tail series for adequate links, a class that generalizes alternating links. We will show that for every adequate link there is a prime alternating link with coinciding tail function. Furthermore, we will strengthen a theorem of Futer, Kalfagianni and Purcell [FKP13, Fut13] and relate the complete tail function of the colored Jones polynomial of an adequate knot to the fiberedness of the link with fiber surface a certain spanning surface of that link.
Acknowledgment: The authors thank Effie Kalfagianni for helpful suggestions and discussions during a visit to LSU.
2. The Main Theorem
2.1. The all state surface and the all graph
Let be a link with link diagram with crossings; to each crossing one can assign either of two Kauffman smoothings as in Figure 1.
Thus there are ways, called states, to assign smoothings to the crossings of the diagram. Two of those states are important to us: The all state and the all state, where either only smoothings or only smoothings are assigned to the crossings. States are represented by smoothing diagrams in the plane. Figure 5 shows the all smoothing diagram for a diagram of the mirror image of the knot . Note, that a link can be recovered from its all or all smoothing diagrams.
The all smoothing and allsmoothing diagram naturally lead to two graphs and where the vertices are the circles of the smoothing diagrams and the edges correspond to the smoothed crossing. A link diagram is called adequate (or adequate) if (or ) does not contain a loop, i.e. an edge that connects a vertex to itself. A link is adequate if it admits an adequate as well as a adequate diagram.
It was shown in [Arm13] that the tail of the colored Jones polynomial of an adequate link exists. For alternating knots this was independently shown by Garoufalidis and Le [GL11] and generalized to higher order tails. An approach via link homologies was given by Rozansky [Roz12].
Main Theorem 2.1.
Suppose two adequate link diagrams only differ locally in their all smoothing diagram as in Figure 4
where the three vertical lines represent arcs in three different smoothed circles. Then the tails of the colored Jones polynomials of the corresponding links coincide.
Remark 2.2.
The conditions on the adequate link diagram in the Main Theorem 2.1 imply that the link diagram is not alternating. For from the three circles in the all smoothing depicted in Figure 4 either the left or the right circle has to lie inside the middle circle and the other circle has to lie outside. In the all smoothing of an alternating diagram for a given circle all other circles either lie outside or inside that circle.
The proof of the Main Theorem is given in Section 3. First we will develop a few Corollaries of the Main Theorem.
The first corollary shows that the tail of the colored Jones polynomial of an adequate link is fully determined by the graph . More specifically, a reduced graph is constructed from by replacing all parallel edges, i.e. edges that connect the same two vertices, by a single edge. Figure 6 gives an example. Then:
Corollary 2.3.
For an adequate link with adequate diagram and reduced allgraph the tail of the colored Jones polynomial only depends on . Moreover, for each adequate link there is a prime alternating link such that the tails of the colored Jones polynomials of the two links coincide.
Proof.
First assume that the link is alternating. Then the circles in the all smoothing diagram trace out faces in the diagram of the knot. Thus is a plane graph. In [AD11] it is shown that the reduction from to does not change the tail of the colored Jones polynomial. Furthermore, by Whitney’s classification theorem all embeddings of a planar graph into the plane are related by isomorphisms. Those correspond to mutations in the link diagram. Since the colored Jones polynomial is invariant under mutations (e.g. [Sto06]) the claim holds for alternating links.
If the link diagram is adequate but not alternating then there is a circle in the all smoothing diagram such that on either side of the circle are other circles. Take an innermost circle of that form and divide it into two arcs and . By the Main Theorem the link can be transformed into a link with equal tail by moving all edges coming from crossings inside the circle to and all edges coming from crossings outside the circle to . Figure 5 gives an example. The resulting link is a connected sum of an alternating link (inside the circle) and an adequate link with fewer crossings (outside the circle). Since the colored Jones polynomial is multiplicative under connected sum the first claim follows. In particular a tail function of the colored Jones polynomial of an adequate link can be expressed as the product of tail functions of colored Jones polynomials of alternating links. As shown in [AD11] the product of the tail functions of two prime alternating links is again the tail function of a prime alternating link and the second claim follows. ∎
Example 2.4.
Figure 5 illustrates the operation in the proof of Corollary 2.3 for the all smoothing diagram of the mirror image of the knot .
In particular, Figure 6 shows that the tail function of the knot is equivalent to the tail function of the connected sum of two (negative) trefoils. It was shown in [AD11] that those tails are express as an evaluation of the two variable Ramanujan theta function:
More specifically: The tail function of the (negative) trefoil is
Thus the tail function of the knot is .
The main theorem immediately implies the following extension of a theorem of Purcell, Kalfagianni and Futer [FKP13, Fut13]. For every state one can construct a spanning surface for the link similar to Seiferts construction of Seifert surfaces (e.g. [Fut13]). Let be the state surface for the all state and as constructed above.
Theorem 2.5.
For a link with diagram and state surface determined by the all state the following statements are equivalent:

fibers over , with fiber .

The reduced all graph is a tree.

The diagram is adequate and .

The diagram is adequate and the tail of the colored Jones polynomial is .
Proof.
Moreover, it directly follows from Corollary 2.3
Corollary 2.6 ([Arm11]).
For a closed positive braid the tail of the colored Jones polynomial is identically .
3. Proof of Main Theorem
3.1. Skein Theory
The Kauffman bracket skein module, , of a manifold and ring with invertible element , is the free module generated by isotopy classes of framed links in , modulo the submodule generated by the Kauffman relations:
If has designated points on the boundary, then the framed links must include arcs which meet all of the designated points.
In this paper we will take , the field of rational functions in variable with coefficients in . As we are concerned with the lowest terms of a polynomial, we will need to express rational functions as Laurent series and define:
Definition 3.1.
Let , define to be the minimum degree of expressed as a Laurent series in .
Note that can be calculated without referring to the Laurent series. Any rational function expressed as where and are both polynomials. Then .
Definition 3.2.
For two Laurent series and we define
if after multiplying by and by , and some powers, to get power series and each with positive constant term, and agree .
We will be concerned with two particular skein modules: , which is isomorphic to under the isomorphism sending the empty link to , and , where has designated points on the boundary. With these designated points, is also called the TemperleyLieb algebra .
We will give an alternate explanation for the TemperleyLieb algebra. First, consider the disk as a rectangle with designated points on the top and designated points on the bottom. Let be the set of all crossingless matchings on these points, and define the product of two crossingless matchings by placing one rectangle on top of the other and deleting any components which do not meet the boundary of the disk. With this product, is a monoid, which we shall call the TemperleyLieb monoid.
Any element in has the form , where . Multiplication in is slightly different from multiplication in , because in complete circles are removed, but in when a circle is removed, it is replaced with .
There is a special element in of fundamental importance to the colored Jones polynomial, called the JonesWentzl idempotent, denoted . Diagramatically this element is represented by an empty box with strands coming out of it on two opposite sides. By convention an next to a strand in a diagram indicates that the strand is replaced by parallel ones.
With
and the JonesWenzl idempotent satisfies
with the properties
The (unreduced) colored Jones polynomial of a link diagram can be defined as the value of the skein relation applied to a diagram , where every component is decorated by an together with the JonesWenzl idempotent. Recall that . To obtain the reduced colored Jones polynomial of a link with diagram , we must compensate for the writhe of and divide by the value on the unknot. That is
We can now properly define the tail of the colored Jones polynomial:
Definition 3.3.
The tail of the reduced colored Jones polynomial of a link – if it exists – is a series , with
The tail of the unreduced colored Jones polynomial of a link – if it exists – is a series , with
In [AD11] and [Arm13] we showed that the (unreduced) tail of adequate links are determined by a collection of crossingless skein diagrams coming from the Bstate graph. Given a link diagram , construct a skein diagram by replacing the former crossings in the all smoothing of by an idempotent colored connecting the two circles.
In [Arm13], the first author found a lower bound for the minimum degree of any element of which contains the JonesWenzl idempotent. Before we explain this bound, consider a crossingless diagram in the plane consisting of arcs connecting JonesWenzl idempotents. We will define what it means for such a diagram to be adequate in much the same way that a knot diagram can be  or adequate.
Construct a crossingless diagram from by replacing each of the JonesWenzl idempotents in by the identity of . Thus is a collection of circles with no crossings. Consider the regions in where the idempotents had previously been. is adequate if no circle in passes through any one of these regions more than once. Figure (a)a shows an example of a diagram that is adequate and Figure (b)b shows an example of a diagram that is not adequate. In both figures every arc is labelled .
If is adequate, then the number of circles in is a local maximum, in the sense that if the idempotents in are replaced by other elements of such that there is exactly one hook total in all of the replacements, then the number of circles in this diagram is one less than the number of circles in . This is the key fact in the proof of the following lemma.
Lemma 3.5 ([Arm13]).
If is expressed as a single crossingless diagram containing JonesWenzl idempotents, then .
If the diagram for is also an adequate diagram, then .
3.2. Proof
We now prove the Main Theorem by using Lemma 3.4 and showing that if two adequate link diagrams and have all smoothings that differ as in Main Theorem 2.1, then . First, an important lemma:
Lemma 3.6.
Proof.
First we use the idempotent property to create a smaller idempotent on strands.
Now we use the recursive relation on the larger idempotent.
If , notice that when applying the recursive relation again on the rightmost term, one of the two terms in the relation will be zero.
Thus we get a simplification:
By performing the recursion a total of times where , we get the following equation:
Letting , the lemma follows.∎
Lemma 3.7.
Given an element of with points on the boundary colored such that the pairings of the with the left and right hand sides of the following equation are adequate diagrams in the plane, then we get the equation
where this equation and the equations appearing in the following proof are actually equations between the pairing of each term with the element . That is the diagrams in this lemma are local pictures of diagrams in .
Remark 3.8.
The abuse of notation in this lemma is unambiguous because if each term is viewed as being an element of , then the equations do not make sense as the equivalence only applies to Laurent polynomials and Laurent series which elements of with colored points on the boundary are not. Thus the equations only make sense if the terms actually represent elements of .
Proof.
For any with , cosider the relation coming from Lemma 3.6 with and :
Note that the two pictures on the left, call them and , are both adequate diagrams and that their minimum degrees are the same by Lemma 3.5. Now we need to compare this with the minimum degree of the rightmost term, call it . First note the number of circles in are fewer than the number of circles in . This is because the diagrams and differ in only one spot where strands running straight across in are replaced by a diagram as in Figure 9.
Because is adequate, the different strands in Figure 9 are closed to form different circles. However, the corresponding strands in are merged in such a way that they close to form a single circle. Thus
Also note that
Finally, note that because , we have . Therefore:
Now by induction on , this completes the proof.∎
The final step to prove Main Theorem 2.1 is to verify the claim mentioned at the beginning of the section that if two adequate link diagrams and have all smoothings that differ as in Main Theorem 2.1, then . This follows from Lemma 3.7. Denote the figure on the right of the equation in Lemma 3.7 because it matches the diagram on the right of the equation in Main Theorem 2.1. To get an expression for we can reflect all diagrams in Lemma 3.7 horizontally. Because the diagram on the left of the equation is symmetric, this shows that , and Main Theorem 2.1 follows.
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